DOCSLIB.ORG

Chapter 1 Post Correspondence Problem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 1 Post Correspondence Problem Contents 1 Post correspondence problem 1 1.1 Definition of the problem ....................................... 1 1.1.1 Alternative definition ..................................... 1 1.2 Example instances of the problem .................................. 1 1.2.1 Example 1 ........................................... 1 1.2.2 Example 2 ........................................... 1 1.3 Proof sketch of undecidability .................................... 2 1.4 Variants ................................................ 2 1.5 References .............................................. 3 1.6 External links ............................................. 3 2 Halting problem 4 2.1 Background .............................................. 4 2.2 Importance and consequences .................................... 4 2.3 Representation as a set ........................................ 5 2.4 Sketch of proof ............................................ 5 2.5 Proof as a corollary of the uncomputability of Kolmogorov complexity ............... 6 2.6 Common pitfalls ........................................... 6 2.7 Formalization ............................................. 7 2.8 Relationship with Gödel’s incompleteness theorems ......................... 7 2.9 Variants of the halting problem .................................... 7 2.9.1 Halting on all inputs ..................................... 8 2.9.2 Recognizing partial solutions ................................. 8 2.9.3 Generalized to oracle machines ............................... 8 2.10 History ................................................ 8 2.11 Avoiding the halting problem ..................................... 9 2.12 See also ................................................ 9 2.13 Notes ................................................. 9 2.14 References .............................................. 9 2.15 External links ............................................. 11 2.16 Text and image sources, contributors, and licenses .......................... 12 2.16.1 Text .............................................. 12 2.16.2 Images ............................................ 12 i ii CONTENTS 2.16.3 Content license ........................................ 12 Chapter 1 Post correspondence problem Not to be confused with the other Post’s problem on the existence of incomparable r.e. Turing degrees. Not to be confused with PCP theorem. g(w) = h(w) The Post correspondence problem is an undecidable 1.2 Example instances of the prob- decision problem that was introduced by Emil Post in 1946.[1] Because it is simpler than the halting problem lem and the Entscheidungsproblem it is often used in proofs of undecidability. 1.2.1 Example 1 Consider the following two lists: 1.1 Definition of the problem A solution to this problem would be the sequence (3, 2, 3, 1), because The input of the problem consists of two finite lists α1; : : : ; αN and β1; : : : ; βN of words over some alphabet A having at least two symbols. A solution to this prob- α3α2α3α1 = bba+ab+bba+a = bbaabbbaa = bb+aa+bb+baa = β3β2β3β1: lem is a sequence of indices (ik)1≤k≤K with K ≥ 1 and 1 ≤ ik ≤ N for all k , such that Furthermore, since (3, 2, 3, 1) is a solution, so are all of its “repetitions”, such as (3, 2, 3, 1, 3, 2, 3, 1), etc.; that is, when a solution exists, there are infinitely many solutions of this repetitive kind. αi1 : : : αiK = βi1 : : : βiK : However, if the two lists had consisted of only α2; α3 and The decision problem then is to decide whether such a β2; β3 from those sets, then there would have been no solution exists or not. solution (the last letter of any such α string is not the same as the letter before it, whereas β only constructs pairs of 1.1.1 Alternative definition the same letter). A convenient way to view an instance of a Post correspon- In the above definition, a solution can be seen as a dence problem is as a collection of blocks of the form f g nonempty word on the alphabet 1;:::;N , and each there being an unlimited supply of each type of block. of the two given lists of words on alphabet A can be Thus the above example is viewed as seen as defining a homomorphism that maps words on f1;:::;Ng to words on A : where the solver has an endless supply of each of these three block types. A solution corresponds to some way of laying blocks next to each other so that the string in the top cells corresponds to the string in the bottom cells. g :(i ; : : : ; i ) 7! α : : : α 1 K i1 iK Then the solution to the above example corresponds to: 7! h :(i1; : : : ; iK ) βi1 : : : βiK : This gives rise to an equivalent alternative definition often 1.2.2 Example 2 found in the literature, according to which any two homo- morphisms g; h with a common domain and a common Again using blocks to represent an instance of the prob- codomain form an instance of the Post correspondence lem, the following is an example that has infinitely many problem, which now asks whether there exists a nonempty solutions in addition to the kind obtained by merely “re- word w in the domain such that peating” a solution. 1 2 CHAPTER 1. POST CORRESPONDENCE PROBLEM In this instance, every sequence of the form (1, 2, 2, . ., the finite state changes, and what happens to the surround- 2, 3) is a solution (in addition to all their repetitions): ing symbols. For example, here the tape head is over a 0 in state 4, and then writes a 1 and moves right, changing to state 7: 1.3 Proof sketch of undecidability Finally, when the top reaches an accepting state, the bot- tom needs a chance to finally catch up to complete the The most common proof for the undecidability of PCP match. To allow this, we extend the computation so that describes an instance of PCP that can simulate the com- once an accepting state is reached, each subsequent ma- putation of a Turing machine on a particular input. A chine step will cause a symbol near the tape head to van- match will occur if and only if the input would be ac- ish, one at a time, until none remain. If qf is an accepting cepted by the Turing machine. Because deciding if a Tur- state, we can represent this with the following transition ing machine will accept an input is a basic undecidable blocks, where a is a tape alphabet symbol: problem, PCP cannot be decidable either. The following There are a number of details to work out, such as dealing discussion is based on Michael Sipser's textbook Intro- with boundaries between states, making sure that our ini- [2] duction to the Theory of Computation. tial tile goes first in the match, and so on, but this shows In more detail, the idea is that the string along the top the general idea of how a static tile puzzle can simulate a and bottom will be a computation history of the Turing Turing machine computation. machine’s computation. This means it will list a string de- The previous example scribing the initial state, followed by a string describing the next state, and so on until it ends with a string describ- q0101#1q401#11q21#1q810. ing an accepting state. The state strings are separated by is represented as the following solution to the Post corre- some separator symbol (usually written #). According to spondence problem: the definition of a Turing machine, the full state of the machine consists of three parts: 1.4 Variants • The current contents of the tape. Many variants of PCP have been considered. One reason • The current state of the finite state machine which is that, when one tries to prove undecidability of some operates the tape head. new problem by reducing from PCP, it often happens that • The current position of the tape head on the tape. the first reduction one finds is not from PCP itself but from an apparently weaker version. Although the tape has infinitely many cells, only some fi- nite prefix of these will be non-blank. We write these • The problem may be phrased in terms of monoid ∗ down as part of our state. To describe the state of the fi- morphisms f, g from the free monoid B to the free ∗ nite control, we create new symbols, labelled q1 through monoid A where B is of size n. The problem is to qk, for each of the finite state machine’s k states. We determine whether there is a word w in B+ such that insert the correct symbol into the string describing the f(w) = g(w).[3] tape’s contents at the position of the tape head, thereby • The condition that the alphabet A have at least two indicating both the tape head’s position and the current symbols is required since the problem is decidable state of the finite control. For the alphabet {0,1}, a typi- if A has only one symbol. cal state might look something like: • A simple variant is to fix n, the number of tiles. This 101101110q700110. problem is decidable if n ≤ 2, but remains undecid- A simple computation history would then look something able for n ≥ 5. It is unknown whether the problem like this: is decidable for 3 ≤ n ≤ 4.[4] q 101#1q 01#11q 1#1q 10. 0 4 2 8 • The circular Post correspondence problem asks We start out with this block, where x is the input string whether indexes i1; i2;::: can be found such that and q0 is the start state: ··· ··· αi1 αik and βi1 βik are conjugate words, i.e., The top starts out “lagging” the bottom by one state, and they are equal modulo rotation. This variant is [5] keeps this lag until the very end stage. Next, for each undecidable. symbol a in the tape alphabet, as well as #, we have a • One of the most important variants of PCP is the “copy” block, which copies it unmodified from one state bounded Post correspondence problem, which to the next: asks if we can find a match using no more than k We also have a block for each position transition the ma- tiles, including repeated tiles. A brute force search chine can make, showing how the tape head moves, how solves the problem in time O(2k), but this may be 1.6.
Load more

Recommended publications
  • A Proof of Cantor's Theorem
    Cantor’s Theorem Joe Roussos 1 Preliminary ideas Two sets have the same number of elements (are equinumerous, or have the same cardinality) iff there is a bijection between the two sets. Mappings: A mapping, or function, is a rule that associates elements of one set with elements of another set. We write this f : X ! Y , f is called the function/mapping, the set X is called the domain, and Y is called the codomain. We specify what the rule is by writing f(x) = y or f : x 7! y. e.g. X = f1; 2; 3g;Y = f2; 4; 6g, the map f(x) = 2x associates each element x 2 X with the element in Y that is double it. A bijection is a mapping that is injective and surjective.1 • Injective (one-to-one): A function is injective if it takes each element of the do- main onto at most one element of the codomain. It never maps more than one element in the domain onto the same element in the codomain. Formally, if f is a function between set X and set Y , then f is injective iff 8a; b 2 X; f(a) = f(b) ! a = b • Surjective (onto): A function is surjective if it maps something onto every element of the codomain. It can map more than one thing onto the same element in the codomain, but it needs to hit everything in the codomain. Formally, if f is a function between set X and set Y , then f is surjective iff 8y 2 Y; 9x 2 X; f(x) = y Figure 1: Injective map.
    [Show full text]
  • Linear Transformation (Sections 1.8, 1.9) General View: Given an Input, the Transformation Produces an Output
    Linear Transformation (Sections 1.8, 1.9) General view: Given an input, the transformation produces an output. In this sense, a function is also a transformation. 1 4 3 1 3 Example. Let A = and x = 1 . Describe matrix-vector multiplication Ax 2 0 5 1 1 1 in the language of transformation. 1 4 3 1 31 5 Ax b 2 0 5 11 8 1 Vector x is transformed into vector b by left matrix multiplication Definition and terminologies. Transformation (or function or mapping) T from Rn to Rm is a rule that assigns to each vector x in Rn a vector T(x) in Rm. • Notation: T: Rn → Rm • Rn is the domain of T • Rm is the codomain of T • T(x) is the image of vector x • The set of all images T(x) is the range of T • When T(x) = Ax, A is a m×n size matrix. Range of T = Span{ column vectors of A} (HW1.8.7) See class notes for other examples. Linear Transformation --- Existence and Uniqueness Questions (Section 1.9) Definition 1: T: Rn → Rm is onto if each b in Rm is the image of at least one x in Rn. • i.e. codomain Rm = range of T • When solve T(x) = b for x (or Ax=b, A is the standard matrix), there exists at least one solution (Existence question). Definition 2: T: Rn → Rm is one-to-one if each b in Rm is the image of at most one x in Rn. • i.e. When solve T(x) = b for x (or Ax=b, A is the standard matrix), there exists either a unique solution or none at all (Uniqueness question).
    [Show full text]
  • Functions and Inverses
    Functions and Inverses CS 2800: Discrete Structures, Spring 2015 Sid Chaudhuri Recap: Relations and Functions ● A relation between sets A !the domain) and B !the codomain" is a set of ordered pairs (a, b) such that a ∈ A, b ∈ B !i.e. it is a subset o# A × B" Cartesian product – The relation maps each a to the corresponding b ● Neither all possible a%s, nor all possible b%s, need be covered – Can be one-one, one&'an(, man(&one, man(&man( Alice CS 2800 Bob A Carol CS 2110 B David CS 3110 Recap: Relations and Functions ● ) function is a relation that maps each element of A to a single element of B – Can be one-one or man(&one – )ll elements o# A must be covered, though not necessaril( all elements o# B – Subset o# B covered b( the #unction is its range/image Alice Balch Bob A Carol Jameson B David Mews Recap: Relations and Functions ● Instead of writing the #unction f as a set of pairs, e usually speci#y its domain and codomain as: f : A → B * and the mapping via a rule such as: f (Heads) = 0.5, f (Tails) = 0.5 or f : x ↦ x2 +he function f maps x to x2 Recap: Relations and Functions ● Instead of writing the #unction f as a set of pairs, e usually speci#y its domain and codomain as: f : A → B * and the mapping via a rule such as: f (Heads) = 0.5, f (Tails) = 0.5 2 or f : x ↦ x f(x) ● Note: the function is f, not f(x), – f(x) is the value assigned b( f the #unction f to input x x Recap: Injectivity ● ) function is injective (one-to-one) if every element in the domain has a unique i'age in the codomain – +hat is, f(x) = f(y) implies x = y Albany NY New York A MA Sacramento B CA Boston ..
    [Show full text]
  • Math 101 B-Packet
    Math 101 B-Packet Scott Rome Winter 2012-13 1 Redefining functions This quarter we have defined a function as a rule which assigns exactly one output to each input, and so far we have been happy with this definition. Unfortunately, this way of thinking of a function is insufficient as things become more complicated in mathematics. For a better understanding of a function, we will first need to define it better. Definition 1.1. Let X; Y be any sets. A function f : X ! Y is a rule which assigns every element of X to an element of Y . The sets X and Y are called the domain and codomain of f respectively. x f(x) y Figure 1: This function f : X ! Y maps x 7! f(x). The green circle indicates the range of the function. Notice y is in the codomain, but f does not map to it. Remark 1.2. It is necessary to define the rule, the domain, and the codomain to define a function. Thus far in the class, we have been \sloppy" when working with functions. Remark 1.3. Notice how in the definition, the function is defined by three things: the rule, the domain, and the codomain. That means you can define functions that seem to be the same, but are actually different as we will see. The domain of a function can be thought of as the set of all inputs (that is, everything in the domain will be mapped somewhere by the function). On the other hand, the codomain of a function is the set of all possible outputs, and a function may not necessarily map to every element of the codomain.
    [Show full text]
  • Sets, Functions, and Domains
    Sets, Functions, and Domains Functions are fundamental to denotational semantics. This chapter introduces functions through set theory, which provides a precise yet intuitive formulation. In addition, the con- cepts of set theory form a foundation for the theory of semantic domains, the value spaces used for giving meaning to languages. We examine the basic principles of sets, functions, and domains in turn. 2.1 SETS __________________________________________________________________________________________________________________________________ A set is a collection; it can contain numbers, persons, other sets, or (almost) anything one wishes. Most of the examples in this book use numbers and sets of numbers as the members of sets. Like any concept, a set needs a representation so that it can be written down. Braces are used to enclose the members of a set. Thus, { 1, 4, 7 } represents the set containing the numbers 1, 4, and 7. These are also sets: { 1, { 1, 4, 7 }, 4 } { red, yellow, grey } {} The last example is the empty set, the set with no members, also written as ∅. When a set has a large number of members, it is more convenient to specify the condi- tions for membership than to write all the members. A set S can be de®ned by S= { x | P(x) }, which says that an object a belongs to S iff (if and only if) a has property P, that is, P(a) holds true. For example, let P be the property ``is an even integer.'' Then { x | x is an even integer } de®nes the set of even integers, an in®nite set. Note that ∅ can be de®ned as the set { x | x≠x }.
    [Show full text]
  • Parametrised Enumeration
    Parametrised Enumeration Arne Meier Februar 2020 color guide (the dark blue is the same with the Gottfried Wilhelm Leibnizuniversity Universität logo) Hannover c: 100 m: 70 Institut für y: 0 k: 0 Theoretische c: 35 m: 0 y: 10 Informatik k: 0 c: 0 m: 0 y: 0 k: 35 Fakultät für Elektrotechnik und Informatik Institut für Theoretische Informatik Fachgebiet TheoretischeWednesday, February 18, 15 Informatik Habilitationsschrift Parametrised Enumeration Arne Meier geboren am 6. Mai 1982 in Hannover Februar 2020 Gutachter Till Tantau Institut für Theoretische Informatik Universität zu Lübeck Gutachter Stefan Woltran Institut für Logic and Computation 192-02 Technische Universität Wien Arne Meier Parametrised Enumeration Habilitationsschrift, Datum der Annahme: 20.11.2019 Gutachter: Till Tantau, Stefan Woltran Gottfried Wilhelm Leibniz Universität Hannover Fachgebiet Theoretische Informatik Institut für Theoretische Informatik Fakultät für Elektrotechnik und Informatik Appelstrasse 4 30167 Hannover Dieses Werk ist lizenziert unter einer Creative Commons “Namensnennung-Nicht kommerziell 3.0 Deutschland” Lizenz. Für Julia, Jonas Heinrich und Leonie Anna. Ihr seid mein größtes Glück auf Erden. Danke für eure Geduld, euer Verständnis und eure Unterstützung. Euer Rückhalt bedeutet mir sehr viel. ♥ If I had an hour to solve a problem, I’d spend 55 minutes thinking about the problem and 5 min- “ utes thinking about solutions. ” — Albert Einstein Abstract In this thesis, we develop a framework of parametrised enumeration complexity. At first, we provide the reader with preliminary notions such as machine models and complexity classes besides proving them to be well-chosen. Then, we study the interplay and the landscape of these classes and present connections to classical enumeration classes.
    [Show full text]
  • Categories and Computability
    Categories and Computability J.R.B. Cockett ∗ Department of Computer Science, University of Calgary, Calgary, T2N 1N4, Alberta, Canada February 22, 2010 Contents 1 Introduction 3 1.1 Categories and foundations . ........ 3 1.2 Categoriesandcomputerscience . ......... 4 1.3 Categoriesandcomputability . ......... 4 I Basic Categories 6 2 Categories and examples 7 2.1 Thedefinitionofacategory . ....... 7 2.1.1 Categories as graphs with composition . ......... 7 2.1.2 Categories as partial semigroups . ........ 8 2.1.3 Categoriesasenrichedcategories . ......... 9 2.1.4 The opposite category and duality . ....... 9 2.2 Examplesofcategories. ....... 10 2.2.1 Preorders ..................................... 10 2.2.2 Finitecategories .............................. 11 2.2.3 Categoriesenrichedinfinitesets . ........ 11 2.2.4 Monoids....................................... 13 2.2.5 Pathcategories................................ 13 2.2.6 Matricesoverarig .............................. 13 2.2.7 Kleenecategories.............................. 14 2.2.8 Sets ......................................... 15 2.2.9 Programminglanguages . 15 2.2.10 Productsandsumsofcategories . ....... 16 2.2.11 Slicecategories .............................. ..... 16 ∗Partially supported by NSERC, Canada. 1 3 Elements of category theory 18 3.1 Epics, monics, retractions, and sections . ............. 18 3.2 Functors........................................ 20 3.3 Natural transformations . ........ 23 3.4 Adjoints........................................ 25 3.4.1 Theuniversalproperty.
    [Show full text]
  • The Entscheidungsproblem and Alan Turing
    The Entscheidungsproblem and Alan Turing Author: Laurel Brodkorb Advisor: Dr. Rachel Epstein Georgia College and State University December 18, 2019 1 Abstract Computability Theory is a branch of mathematics that was developed by Alonzo Church, Kurt G¨odel,and Alan Turing during the 1930s. This paper explores their work to formally define what it means for something to be computable. Most importantly, this paper gives an in-depth look at Turing's 1936 paper, \On Computable Numbers, with an Application to the Entscheidungsproblem." It further explores Turing's life and impact. 2 Historic Background Before 1930, not much was defined about computability theory because it was an unexplored field (Soare 4). From the late 1930s to the early 1940s, mathematicians worked to develop formal definitions of computable functions and sets, and applying those definitions to solve logic problems, but these problems were not new. It was not until the 1800s that mathematicians began to set an axiomatic system of math. This created problems like how to define computability. In the 1800s, Cantor defined what we call today \naive set theory" which was inconsistent. During the height of his mathematical period, Hilbert defended Cantor but suggested an axiomatic system be established, and characterized the Entscheidungsproblem as the \fundamental problem of mathemat- ical logic" (Soare 227). The Entscheidungsproblem was proposed by David Hilbert and Wilhelm Ackerman in 1928. The Entscheidungsproblem, or Decision Problem, states that given all the axioms of math, there is an algorithm that can tell if a proposition is provable. During the 1930s, Alonzo Church, Stephen Kleene, Kurt G¨odel,and Alan Turing worked to formalize how we compute anything from real numbers to sets of functions to solve the Entscheidungsproblem.
    [Show full text]
  • Consider the “Function” F(X) = 1 X − 4 . in Fact, This Is Not a Function Until I Have Told You It's Domain and Codomain
    Consider the “function” 1 f(x) = . x − 4 In fact, this is not a function until I have told you it’s domain and codomain. More accurately we have to talk about the function 1 f : R \{4} → R, f(x) = . x − 4 It’s important to get in the habit of specifying the domain and codomain. In fact the definition of function f : A → B specifies it as a subset of the cartesian produce A × B. From this point of view, our function is the set {(x, 1/(x − 4) : x ∈ R \{4}} ⊂ R \{4} × R 1 Note: my definition of function is different from that of the book. Book does not require that the function be defined everywhere in A. Range The range of a function f : A → B is the subset of B consisting of values of the function. A function is surjective (or onto) if the range equals the codomain. The function f : R \{4} → R given by f(x) = 1/(x − 4) is not surjective because it misses the value 0. That is, the equation g(x) = 0 has no solution in R \{4}. 2 However, if we define a new function g : R \{4} → R \{0}, using the same formula g(x) = 1/(x−4), then g is a surjective function. Note that although the formula is the same, this is a different function from f because we have changed the codomain. To prove g is surjective, we must show that if y ∈ R \{0}, then the equation g(x) = y has a solution x ∈ R\{4}.
    [Show full text]
  • SET THEORY for CATEGORY THEORY 3 the Category Is Well-Powered, Meaning That Each Object Has Only a Set of Iso- Morphism Classes of Subobjects
    SET THEORY FOR CATEGORY THEORY MICHAEL A. SHULMAN Abstract. Questions of set-theoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made can have noticeable effects on what categorical constructions are permissible. In this expository paper we summarize and compare a num- ber of such “set-theoretic foundations for category theory,” and describe their implications for the everyday use of category theory. We assume the reader has some basic knowledge of category theory, but little or no prior experience with formal logic or set theory. 1. Introduction Since its earliest days, category theory has had to deal with set-theoretic ques- tions. This is because unlike in most fields of mathematics outside of set theory, questions of size play an essential role in category theory. A good example is Freyd’s Special Adjoint Functor Theorem: a functor from a complete, locally small, and well-powered category with a cogenerating set to a locally small category has a left adjoint if and only if it preserves small limits. This is always one of the first results I quote when people ask me “are there any real theorems in category theory?” So it is all the more striking that it involves in an essential way notions like ‘locally small’, ‘small limits’, and ‘cogenerating set’ which refer explicitly to the difference between sets and proper classes (or between small sets and large sets). Despite this, in my experience there is a certain amount of confusion among users and students of category theory about its foundations, and in particular about what constructions on large categories are or are not possible.
    [Show full text]
  • The Mereological Foundation of Megethology
    The Mereological foundation of Megethology Massimiliano Carrara, Enrico Martino FISPPA Department, University of Padua (Italy) April 24, 2015 Abstract In Mathematics is megethology (Lewis, 1993) David K. Lewis pro- poses a structuralist reconstruction of classical set theory based on mereology. In order to formulate suitable hypotheses about the size of the universe of individuals without the help of set-theoretical notions, he uses the device of Boolos' plural quantification for treating second order logic without commitment to set-theoretical entities. In this paper we show how, assuming the existence of a pairing function on atoms, as the unique assumption non expressed in a mere- ological language, a mereological foundation of set theory is achievable within first order logic. Furthermore, we show how a mereological cod- ification of ordered pairs is achievable with a very restricted use of the notion of plurality without plural quantification. 1 1 Introduction David Lewis (Lewis, 1993) proposes a reconstruction of classical set theory grounded on mereology and plural quantification (see also (Lewis, 1991)).1 Mereology is the theory of parthood relation. One of its first formaliza- tions is Leonard and Goodman Calculus of individuals ((Leonard & Good- man, 1940), 33-40).2 Plural quantification is a reinterpretation of second order monadic logic, proposed by Boolos in (Boolos, 1984), (Boolos, 1985).3 In Boolos' perspective second order monadic logic is ontologically innocent: contrary to the most accredited view, it doesn't entail any commitment to classes or to properties but only to individuals, as first order logic does. Acccording to Boolos, second order quantification differs from first order quantification only in that it refers to individuals plurally, while the latter refers to individuals singularly.
    [Show full text]
  • Sets, Relations, Functions
    Part I Sets, Relations, Functions 1 The material in this part is a reasonably complete introduction to basic naive set theory. Unless students can be assumed to have this background, it's probably advisable to start a course with a review of this material, at least the part on sets, functions, and relations. This should ensure that all students have the basic facility with mathematical notation required for any of the other logical sections. NB: This part does not cover induction directly. The presentation here would benefit from additional examples, espe- cially, \real life" examples of relations of interest to the audience. It is planned to expand this part to cover naive set theory more exten- sively. 2 sets-functions-relations rev: 41837d3 (2019-08-27) by OLP/ CC{BY Chapter 1 Sets 1.1 Basics sfr:set:bas: Sets are the most fundamental building blocks of mathematical objects. In fact, explanation sec almost every mathematical object can be seen as a set of some kind. In logic, as in other parts of mathematics, sets and set-theoretical talk is ubiquitous. So it will be important to discuss what sets are, and introduce the notations necessary to talk about sets and operations on sets in a standard way. Definition 1.1 (Set). A set is a collection of objects, considered independently of the way it is specified, of the order of the objects in the set, or of their multiplicity. The objects making up the set are called elements or members of the set. If a is an element of a set X, we write a 2 X (otherwise, a2 = X).
    [Show full text]